It's not about playing through a painful injury. All NBA players will do that and all of them take pain killers for this anyway. It's about whether or not there is a risk of further injury. With a broken finger once it's padded and strapped you aren't at a high risk to further injure it. With a freshly stitched cut there's a risk of popping the stitches which can cause infections and cause you to miss even more games with more a serious injury